Information projection

In information theory, the information projection or I-projection of a probability distribution q onto a set of distributions P is

p^{*}={\underset {p\in P}{\arg \min }}\operatorname {D}_{{{\mathrm {KL}}}}(p||q)

where $D_{{{\mathrm {KL}}}}$ is the Kullback–Leibler divergence from p to q. Viewing the Kullback–Leibler divergence as a measure of distance, the I-projection $p^{*}$ is the "closest" distribution to q of all the distributions in P.

The I-projection is useful in setting up information geometry, notably because of the following inequality:^[1]

$\operatorname {D} _{\mathrm {KL} }(p||q)\geq \operatorname {D} _{\mathrm {KL} }(p||p^{*})+\operatorname {D} _{\mathrm {KL} }(p^{*}||q)$

This inequality can be interpreted as an information-geometric version of Pythagoras' triangle inequality theorem, where KL divergence is viewed as squared distance in a Euclidean space.

It is worthwile to note that since $\operatorname {D} _{\mathrm {KL} }(p||q)\geq 0$ and continuous in p, if P is closed and non-empty, then there exists at least one minimizer to the optimization problem framed above. Furthermore if P is convex, then the optimum distribution is unique.

The reverse I-projection is

$p^{*}={\underset {p\in P}{\arg \min }}\operatorname {D} _{\mathrm {KL} }(q||p)$

References

↑ Cover, Thomas M.; Thomas, Joy A. (2006). Elements of Information Theory (2 ed.). Hoboken, New Jersey: Wiley Interscience. pp. 367(theorem 11.6.1).

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Information projection

See also

References